Robust fault diagnosis of rolling bearings via entropy-weighted nuisance attribute projection and neural network under various operating conditions

Abstract

Rolling bearings are crucial components in the fields of mechanical, civil, and aerospace engineering. They sometimes work under various operating conditions, which makes it harder to distinguish faults from normal signals. Nuisance attribute projection (NAP) is a technique that has been widely used in audio and image recognition to eliminate interference information in the extracted feature space. In constructing the weighted matrix of NAP, the setting of the weighted value represents the degree of interference between the feature vectors. The interference is either taken into consideration in whole, or not considered at all, which will inevitably lead to information loss. In our work, an entropy-weighted NAP (EWNAP) is proposed to deal with such “bipolar problem” in constructing the weighted matrix. The eigenvalues of covariance matrix of collected signals contain dynamical information, and the fuzzy entropy is adopted to evaluate the dispersion degree of these eigenvalues. After normalization, these entropy values are used to express the weight relationship in the weighted matrix of EWNAP. The features processed by EWNAP can be used as samples and combined with neural network to achieve fault diagnosis of rolling bearings. Furthermore, a fault diagnosis approach with insufficient data is demonstrated to validate the effectiveness of the proposed scheme. In the case studies, Case Western Reserve University bearing database and data collected from the bearing fault simulation bench are used. These case studies show that the proposed EWNAP alleviates the interference caused by various operating conditions, and the comparative analysis confirms that the proposed method works better than the conventional methods.

Keywords

Rolling bearing fault diagnosis entropy-weighted nuisance attribute projection fuzzy entropy various operating conditions neural network

Introduction

With the rapid and continuous developments of science, technology, and industrial applications, mechanical equipment is becoming more and more complex. Rolling bearings are widely used in rotating machinery, and its fault diagnosis is crucial in practical applications. It is of great importance to develop an effective and intelligent system for the fault diagnosis of rolling bearings, ensuring their healthy operation. ^1–2 In practice, rolling bearings work under various operating conditions with different rotational speeds and loads. The collected vibration signals of rolling bearings contain a lot of status information besides the dynamical information of rolling bearings. Consequently, the fault characteristics of rolling bearings are usually blended with the status information, and it is hard to extract the fault characteristics under various operating conditions.³ Hence, it is of great significance to extract fault information of rolling bearings among existing redundant information under various operating conditions.

Feature extraction is very important in the field of fault diagnosis of rolling bearings. The extracted features directly affect the practicability of fault diagnosis schemes. The redundant information in the collected vibration signals makes it difficult to identify fault characteristics of faulty rolling bearings. To tackle this problem, several methods have been proposed including linear discriminant analysis (LDA) ⁴ and principal component analysis (PCA).⁵ LDA keeps the maximum inter-class and minimum inter-class properties of datasets during linear projection, by finding the optimal projection axes. PCA distinguishes the global maximum variance of datasets to extract the key features by projection. In general, LDA and PCA can extract sensitive features of the vibration signals of faulty components by decreasing the dimensionality of feature vectors, and they have been adopted in the field of image processing,^6–7 audio recognition,^8–9 and fault diagnosis of rotating machinery.^10–11 In addition, kernel function is adopted to improve the computation efficiency of LDA and PCA, such as kernel linear discriminant analysis (KLDA)¹² and kernel principal component analysis (KPCA).¹³ These two methods are the typical kernel feature extraction methods, using a kernel function to conduct computational process instead of mapping into a high-dimensional feature space. KLDA and KPCA have been widely used in the application of dimensionality reduction for large scale feature extraction.^14–15 The studies show that while LDA and PCA derived approaches are valid dimensionality reduction methods, they suffer from the problems of low-computational efficiency and inaccurate feature extraction of local structure when processing large datasets by only considering structures in whole.¹⁶

Recently, locality preserving projection (LPP) is proposed to explore the nonlinear structure of the manifold of datasets. It extracts essential components of the inherent linear and nonlinear manifold structures by retaining its local structure, and this very characteristic has been illustrated in many studies.^17–18 Afterwards, orthogonal locality preserving projection (OLPP) is proposed as an extension to LPP. It reduces the dimension of features by calculating the orthogonal projection vectors, while maintaining local structure information.¹⁹ In general, LPP and OLPP extract essential information of vibration signals by considering local structures of datasets. The superiorities of these methods compared to the conventional methods have been illustrated.^20–21 The interference information from operating conditions is rather difficult to alleviate, which makes it hard to diagnose faults of rolling bearings under various operating conditions.²² To address this issue, nuisance attribute projection (NAP) has been adopted by researchers because it could alleviate the redundant interference caused by multiple channels in the field of speech recognition.²³ The algorithm of NAP is designed to filter out the components of nuisance attributes in the feature space and extract the general characteristic information. It has been extensively applied to speech, image, and face recognitions.^24–26 Recently, NAP has gradually drawn more attention from researchers in the field of fault diagnosis of rotating machinery. Huang et al.²² adopted NAP and ranking mutual information (RMI) to achieve fault diagnosis of health degradation monitoring of rolling bearings. Jiang et al.^27–29 proposed a method by combining NAP and hidden Markov model (HMM) to evaluate the performance degradation of rolling bearings, and then they combined NAP with Student’s t-hidden Markov model (Student’s t-HMM) to obtain more accurate performance degradation assessments of rolling bearings.³⁰ Ma et al.³¹ imported NAP to the structure of convolutional neural network (CNN) to simplify the diagnosis process of rolling bearings under various speed conditions. Gandhi et al.³² introduced NAP to eliminate the interference of system dependent features and improve the robustness of fault diagnosis system of synchronous generators. Further, they proposed a stacked NAP approach to obtain better classification performance of fault diagnosis system of synchronous generators.³³ Gopinath et al.³⁴ used NAP to extract fault features, and then put them into classification and regression trees (CART) to achieve intelligent fault identification of synchronous generators. Meanwhile, they explored locality constrained linear coding (LLC) combined with NAP to construct scalable fault models for diagnosis in synchronous generators.³⁵ The applications of NAP in the fault diagnosis domain are summarized in Table 1.

Table 1.

Summary of applications of nuisance attribute projection in machine fault diagnosis.

Objects	Methodologies	References
Bearings	NAP+RMI	22
Bearings	NAP+HMM	27,28,29
Bearings	NAP+Student’s t-HMM	30
Bearings	NAP+CNN	31
Motors	NAP+Support Vector Machine (SVM)	32
Motors	Stacked NAP+SVM	33
Motors	NAP+CART	34
Motors	NAP+LLC+SVM	35

In the area of fault diagnosis of rolling bearings, various operating conditions can impact fault feature extraction of rolling bearings because of the interference information. The abovementioned analysis of NAP applications shows that NAP can alleviate such interference attributes by increasing the proportion of fault features during projection. During the process of eliminating the interference attributes, NAP inevitably removes some valuable information due to the “bipolar problem” existing in constructing weighted matrix. Such a problem refers to the phenomenon that the interference is either taken into consideration in whole, or not considered at all. To address this problem, an entropy-weighted nuisance attribute projection (EWNAP) algorithm is proposed in this paper work. EWNAP constructs a quantitative weighted matrix during the projection process. The interference degrees are taken into consideration comprehensively and quantitatively estimated. The proposed algorithm is designed to avoid removing essential fault features, and to effectively extract fault features of rolling bearings under various operating conditions.

The rest of this paper is organized as follows. Methodologies illustrates the methodologies of NAP, EWNAP, and back propagation (BP) neural network. Robust Fault Diagnosis of Rolling Bearings via EWNAP and BP Neural Network proposes the new robust fault diagnosis approach combining EWNAP and BP neural network, along with the fault diagnosis scheme with insufficient data. Case Studies verifies the effectiveness and superiority of the proposed approach by several case studies. Conclusion concludes the paper with discussions.

Methodologies

Nuisance attribute projection

Due to the various operating conditions, there exist nuisance attributes in the collected vibration signals of rolling bearings. These nuisance attributes affect the effectiveness and accuracy of fault diagnosis of rolling bearings. For this reason, NAP has been used to alleviate the interference of operating conditions in the feature space to achieve desired diagnosis results. The theory of NAP is illustrated in Figure 1. $x$ denotes the vector containing both fault information and nuisance attributes, $x^{'}$ denotes essential fault characteristics, and $Q$ denotes the subspace containing the nuisance attributes. NAP constructs projection matrix $P$ to determine the subspace to eliminate the nuisance attributes.

Considering a $N$ -dimensional feature space, $m$ samples from different operating conditions can be expressed as a $N \times m$ matrix $X = [x_{1}, x_{2}, \dots, x_{m}]$ . The NAP algorithm can eliminate redundant disturbances in the feature space as follows

X^{'} = P X

(1)

where

P

denotes projection matrix, which is a

N \times N

matrix, and can be expressed as follows

P = I - \sum_{i = 1}^{d} w_{i} w_{i}^{T}

(2)

where

I

denotes an identity matrix of

N \times N

i

denotes the

i

th projection direction, and

w_{i}

is a orthonormal vector.

d

represents the number of projection directions of nuisance components. It needs to be noted that

d

value can be decided corresponding to Ref. 36. A larger

d

leads to better projection results, but results in more computational load, meaning less computation efficiency. Hence, projection effect and computation efficiency are both considered. To eliminate nuisance attributes from original feature space, the projection impact (PI) needs to be minimized

PI = \sum_{i j} W_{i j} {‖ P \cdot x_{i} - P \cdot x_{j} ‖}^{2}

(3)

where

W_{i j}

is positive when

x_{i}

and

x_{j}

are from different operation conditions, and

W_{i j}

is zero when

x_{i}

and

x_{j}

are from same operation condition. The nuisance attributes need to be eliminated under various operation conditions. The elements of weighted matrix

W

can be defined as follows

W_{i j} = {\begin{matrix} 1, if operating condition (x_{i}) \neq operating condition (x_{j}) \\ 0, otherwise \end{matrix}

(4)

The weighted matrix represents the relationships between all vectors. The projection matrix $P$ in equation (3) is calculated to obtain the leading eigenvectors of the following eigenvalue problem

X Z (W) X^{T} w_{i} = λ_{i} w_{i} (i = 1, ..., d)

(5)

where the matrix

Z (W) = d i a g (W \cdot I) - W

, and

d i a g (\cdot)

is an operator to form diagonal matrix from a vector. The projection matrix

P

can be obtained and adopted in the process of eliminating nuisance attributes from the original feature space.

Entropy-weighted nuisance attribute projection

The rolling bearing is affected by speed, load, and other factors during operation. The interference information due to these factors is more complex, and the extent of interference is generally different. However, NAP cannot quantify the extent of interference and inevitably removes some sensitive features. Mateika et al.³⁷ proposed a method for subspace estimation of nuisance existing in different channels. The nuisance attributes of different channels are quantitatively measured via the dispersion degree of eigenvalues of the covariance matrix. In this paper, the idea of the above method is referenced to quantify the interference degree of various operating conditions. Since the dispersion of eigenvalues of the covariance matrix is positively correlated to the amount of disturbances encountered, the more dispersed the eigenvalues of the covariance matrix are, the more disturbance information exists in the current operating condition. On the contrary, the more concentrated the eigenvalues are, the less interference exists in the current operating condition.

First, the eigenvalues of the covariance matrix can be calculated with

(\sum_{k = 1}^{K_{n}} (μ_{n, k} - {\bar{μ}}_{n}) {(μ_{n, k} - {\bar{μ}}_{n})}^{T}) U_{n} = Λ_{n} U_{n}

(6)

where

K_{n}

denotes the number of signal samples of a certain pattern under the

n_{t h}

(n = 1, 2, \dots, N)

operating condition.

μ_{n, k}

denotes root-mean-square (RMS) of the

k_{t h}

sample under the

n_{t h}

operating condition.

{\bar{μ}}_{n}

denotes the mean value of RMS of all

K

signal samples under the

n_{t h}

operating condition.

U_{n}

denotes eigenvector matrix, and all eigenvalues are arranged as

Λ_{n} = {λ_{n, 1}, λ_{n, 1}, \dots, λ_{n, k}, \dots, λ_{n, K}}

In the field of audio signal processing, the sources of nuisance are quantitatively estimated by computing the dispersion degree of eigenvalues of the covariance matrix. In mechanical fault diagnosis, the dispersion degree is hard to define. Hence, entropy is chosen to quantify the dispersion degree in this paper, to account for its internal essence of measuring the regularity and complexity of dynamic systems. Fuzzy entropy is a method based on the concept of approximate entropy and sample entropy to measure the complexity, by defining the concept of membership function.³⁸ Fuzzy entropy employs the least mean square and membership function method to process the similarity measures of sequence. Thus, it can be used as a quantitative index to describe the dispersion degree.³⁹

The procedures of the computation of fuzzy entropy are described in detail in Deng’s paper.⁴⁰ Since the length $Λ_{n}$ is finite, the fuzzy entropy of $Λ_{n}$ can be defined as follows

E_{n} = In ϕ^{m} (K, r) - In ϕ^{m + 1} (K, r)

(7)

where

ϕ^{m}

is the function of fuzzy entropy, and

m

is the model dimension.

r

and

K

are threshold and length of eigenvalue sequence

Λ_{n}

, respectively. In this paper,

E_{n}

stands for the fuzzy entropy value, denoting the dispersion degree of a certain pattern signal of rolling bearing under

n_{t h}

operating condition. The larger

E_{n}

is, the greater the interference is.

E_{n}

is normalized as follows

{\tilde{E}}_{n} = \frac{E_{n} - \min (E_{n})}{\max (E_{n}) - \min (E_{n})}

(8)

The proposed entropy-weighted NAP (EWNAP) by the authors is defined as follows

W_{i j} = {\begin{matrix} \sqrt{{\tilde{E}}_{x_{i}}^{2} + {\tilde{E}}_{x_{j}}^{2}}, if operating condition (x_{i}) \neq operating condition (x_{j}) \\ 0, otherwise \end{matrix}

(9)

when

x_{i}

and

x_{j}

belong to one fault type from various operating conditions, it is necessary to relieve the interference components caused by the operating conditions, and the weighted value is defined as the square root of the square sum of normalized fuzzy entropy values in the two operating conditions. For

x_{i}

and

x_{j}

under the same operating condition, no interference component needs to be considered, and hence the weighted value is defined as 0.

The “bipolar problem” refers to the phenomenon that the interference is either taken into consideration in whole where weighted coefficient is chosen as 1, or not to be considered at all where weighted coefficient is chosen as 0. Hence, the above entropy-weighted scheme is proposed by the authors to eliminate the “bipolar problem”. The EWNAP is illustrated in Figure 2, and the detailed procedure of the algorithm is shown in Figure 3.

Back propagation neural network

BP neural network is a multilayer backward neural network, with the advantages of forward propagation of signals and backward propagation of errors.⁴¹ When output values differ from expected values, it would start to propagate backwards and optimize its topology and network weights by self-learning and self-adjustment.^42–43 In this paper, BP neural network is employed to diagnose rolling bearing faults. BP neural network consists of one input layer, one hidden layer, and one output layer. It adopts nonlinear differential functions in weights training. The input layer receives the training samples and transmits them to the output layer through the hidden layer. In order to reduce the errors, the weights correction is carried out layer by layer from the output layer to the input layer.⁴⁴ The input layer represents the features of four rolling bearing types, namely normal, inner race defect, ball defect, and outer race defect. The output values describe the classification results of four types.

Robust fault diagnosis of rolling bearings via EWNAP and BP neural network

Studies show NAP can alleviate the interference of various operating conditions, which improves the accuracy of fault diagnosis. Entropy measures the complexity and regularity of signals, and its effectiveness has been validated in mechanical fault diagnosis and structural health monitoring.⁴⁵ BP neural network is one of the most popular machine learning methods to achieve fault classification, evaluation, and prediction.⁴⁶ Hence, in this paper, a feature collection is constructed by optimal time and frequency domain features combined with entropy values. Then, fuzzy entropy is used to quantitatively estimate the interference degree of various operating conditions, thus obtaining weighted matrix to solve the “bipolar problem” existed in NAP. EWNAP is proposed to obtain features with alleviating the interference of various operating conditions. Finally, the samples composed of these features are input into BP neural network for fault classification to achieve robust fault diagnosis of rolling bearings.

The main steps of the proposed robust fault diagnosis of rolling bearing via EWNAP and BP neural network approach are as follows:

(1) The optimal 11 time-domain features and 8 frequency-domain features are extracted from vibration signals of rolling bearings.⁴⁷ Furthermore, empirical mode decomposition (EMD) is conducted on these signals.⁴⁸ Since the first 5 to 6 intrinsic mode functions (IMFs) from signals processed by EMD contain the main fault information,^49–50 the sample entropy (SE) values of the first 6 IMFs are computed to extract more valid features.^51–52 These features are listed in Table 2. In this way, vectors with 25-dimension are obtained by the feature extraction process.

(2) After feature extraction, each feature matrix consists of vectors belonging to the same type under a certain operating condition. The normalization is conducted on these feature matrixes, and EWNAP is further applied to these normalized matrixes denoting the same type with different operating conditions. Thus, the projection matrix $P_{m}$ of each type can be acquired. According to the idea of constructing the fusion mapping matrix in Ding’s paper,¹⁷ projection matrixes of all types can constitute a matrix collection called $P_{s}$ and the specific processes are illustrated in Figure 4.

(3) The vectors of normalized feature matrix of each known type are projected with $P_{s}$ , and adopted as training samples for BP neural network. Afterwards, the vectors of normalized feature matrix of unknown type are also projected with $P_{s}$ , and then put into the constructed BP neural network for testing process. Thus, the trained BP neural network by the vectors of normalized feature matrixes provides the fault classification and identification results. The schematic diagram of the proposed robust fault diagnosis method via EWNAP and BP neural network is shown in Figure 5.

(4) To improve the traditional scheme, the scheme via EWNAP and BP neural network with insufficient data are proposed by the authors. The number of feature vectors constituted by extracted features as training samples under one operating condition is reduced, and the feature vectors under the same operating condition are used as testing samples to verify the fault identification ability of the proposed method in cases of insufficient data. The scheme of the proposed method is illustrated in Figure 6.

Table 2.

Feature parameters.

Time-domain feature parameters	Frequency-domain feature parameters	Entropy feature parameters
$F_{1} = \frac{\sum_{n = 1}^{N} x (n)}{N}$	$F_{12} = \frac{\sum_{k = 1}^{K} s (k)}{K}$	$F_{20} = S E (m, r, I M F 1)$
$F_{2} = \sqrt{\frac{\sum_{n = 1}^{N} {(x (n) - F_{1})}^{2}}{N - 1}}$	$F_{13} = \frac{\sum_{k = 1}^{K} {(s (k) - F_{12})}^{2}}{K - 1}$	$F_{21} = S E (m, r, I M F 2)$
$F_{3} = {(\frac{\sum_{n = 1}^{N} \sqrt{\| x (n) \|}}{N})}^{2}$	$F_{14} = \frac{\sum_{k = 1}^{K} {(s (k) - F_{12})}^{3}}{K {(\sqrt{F_{13}})}^{3}}$	$F_{22} = S E (m, r, I M F 3)$
$F_{4} = \sqrt{\frac{{\sum_{n = 1}^{N} (x (n))}^{2}}{N}}$	$F_{15} = \frac{\sum_{k = 1}^{K} f_{k} s (k)}{\sum_{k = 1}^{K} s (k)}$	$F_{23} = S E (m, r, I M F 4)$
$F_{5} = \max \| x (n) \|$	$F_{16} = \sqrt{\frac{\sum_{k = 1}^{K} {(f_{k} - F_{15})}^{2} s (k)}{K}}$	$F_{24} = S E (m, r, I M F 5)$
$F_{6} = \frac{\sum_{n = 1}^{N} {(x (n) - F_{1})}^{3}}{(N - 1) F_{2}^{3}}$	$F_{17} = \frac{\sum_{k = 1}^{K} f_{k}^{2} s (k)}{\sqrt{\sum_{k = 1}^{K} s (k) \sum_{k = 1}^{K} f_{k}^{4} s (k)}}$	$F_{25} = S E (m, r, I M F 6)$
$F_{7} = \frac{\sum_{n = 1}^{N} {(x (n) - F_{1})}^{4}}{(N - 1) F_{2}^{4}}$	$F_{18} = \frac{F_{16}}{F_{15}}$
$F_{8} = \frac{F_{5}}{F_{4}}$	$F_{19} = \frac{\sum_{k = 1}^{K} {(f_{k} - F_{15})}^{1 / 2} s (k)}{K \sqrt{F_{16}}}$
$F_{9} = \frac{F_{5}}{F_{3}}$
$F_{10} = \frac{F_{4}}{\frac{1}{N} \sum_{n = 1}^{N} \| x (n) \|}$
$F_{11} = \frac{F_{5}}{\frac{1}{N} \sum_{n = 1}^{N} \| x (n) \|}$
where $x (n)$ is a signal sequence for $n = 1,2, \dots, N$ .	where $s (k)$ is a spectrum for $k = 1,2, \dots, K$ , $f_{k}$ is the frequency value of the $k^{t h}$ spectrum.	$m$ m and $r$ are the parameters defined in SE.⁴⁹

Case studies

Practically, fault diagnosis of rotating machinery is usually interfered by various operating conditions such as rotating speed and load. It leads to inaccurate results in fault diagnosis, especially in the feature extraction of vibration signals. Three case studies have been investigated to demonstrate how the proposed approach can effectively handle this problem. Case 1 is to demonstrate the effectiveness of EWNAP. Case 2 is to verify the superiority of the proposed approach via EWNAP and BP neural network. Case 3 is to verify the effectiveness of the proposed approach with insufficient data.

Case 1: Processing Case Western Reserve University Bearing Database

Bearing datasets from Case Western Reserve University Bearing Database ⁵³ are adopted to validate the effectiveness of EWNAP. The experiment uses 6205-2RS JEM SKF rolling bearings. Electro discharge machining is used to obtain different fault types of rolling bearings, including inner race, ball, and outer race defects. Three sensors are installed in different locations of the experimental equipment, separately at motor supporting base plate, 12 o’clock positions of fan, and drive ends of the motor housing. The sampling frequency is 12 kHz. The schematic diagram and a picture of the experimental equipment are shown in Figure 7. The analyzed rolling bearing data sets under various operating conditions are illustrated in Table 3. Due to the existence of interferences caused by operating conditions and noises, acquired signals are processed by maximum correlated kurtosis deconvolution to clearly show the presence of faults.⁵⁴ Take rolling bearings under the operating condition of 0 Hp and 1797 rpm as examples. Spectrums and envelope spectrums of acquired signals of rolling bearings are respectively shown in Figure 8 and Figure 9. The representations of different bearing characteristic frequencies are shown in Table 4.

Table 3.

Illustrations of rolling bearing data sets under various operating conditions.

Condition	Type	Samples	Motor load (Hp)	Approx. Motor speed (rpm)
1	Normal	30	0	1797
2			1	1772
3			3	1730
4	Inner race defect	30	0	1797
5			1	1772
6			3	1730
7	Ball defect	30	0	1797
8			1	1772
9			3	1730
10	Outer race defect	30	0	1797
11			1	1772
12			3	1730

Table 4.

The representations of different bearing characteristic frequencies.

Rotational frequency	Defect on inner race	Defect on ball	Defect on outer race
$f_{r}$	$f_{i}$	$f_{b}$	$f_{o}$

During the experiment, the defect diameter of faulty rolling bearing is 0.007 inch. The motor loads are 0, 1, and 3 Hp, and the speeds are 1797, 1772, and 1730 rpm, which are shown in Table 3. For each type of rolling bearing, the experiment is conducted under 3 various operating conditions. Hence there are 12 conditions. For each condition, 30 samples with the length of 4096 points are collected. And for each sample, 11 time-domain features (mean, variance, etc.), 8 frequency-domain features (average amplitude, square root amplitude, etc.), and 6 intrinsic mode entropy values are extracted to construct a vector. Intrinsic mode entropy value denotes sample entropy values of IMF decomposed by EMD. Hence, for each condition, a feature matrix with size of 25 $\times$ × 30 can be obtained. For each type, there are 3 conditions corresponding to 3 feature matrices. In the EWNAP algorithm, it should be noticed that in the algorithm, while a lager $d$ leads to a better projection effect, it also results in more computational load, meaning less computation efficiency.⁵⁵ Through computation, it is found that PI decreases monotonically with the increase of $d$ , and remains stable after $d$ reaches 20. Taking projection effect and computation efficiency into account, the parameter $d$ can be chosen as 20. Then, the processed feature matrix can be obtained by the projection matrix $P_{m}$ of each type. The original features and features processed by EWNAP of a vector under each condition are shown in Figure 10 and Figure 11, respectively.

It can be seen from Figure 10 that there are differences in the feature curves of the same type under various operating conditions before projection. After projection, as shown in Figure 11, the number of features is unchanged, and the feature curves of the same type under various operating conditions are converted to the same curve. The correlation coefficients between two curves of one type are calculated and average correlation coefficients of each type are listed in Table 5.

Table 5.

The mean correlation coefficients of each type under various operating conditions.

	Original	EWNAP
Normal	0.89	0.99
Inner race defect	0.92	1.00
Ball defect	0.91	1.00
Outer race defect	0.93	1.00

It can be found from Table 5 that, compared with the mean correlation coefficient of original feature curves, the mean correlation coefficients of processed features curves by EWNAP are higher than original ones. The average correlation coefficient of each type is equal to 1 or very close to 1 after EWNAP processing. The quantitative analysis shows that EWNAP can effectively eliminate the adverse impact of various speeds and loads, namely various operating conditions. The case study on the Case Western University Bearing Database in this subsection can demonstrate the effectiveness of EWNAP.

Case 2: Validation of robust fault diagnosis via EWNAP and BP neural network

In this section, the collected vibration signals of rolling bearings under various operating conditions are used to test the effect of the proposed fault diagnosis approach via EWNAP and BP neural network. The bearing fault simulation bench is utilized for the second experiment, as shown in Figure 12. The sampling frequency is 2560 Hz. Different fault types of rolling bearings are processed using electron discharge machining technology, and four types of rolling bearing, including normal rolling bearing and faulty rolling bearing with inner race defect, ball defect, and outer race defect, respectively. The photos of four kinds of rolling bearings are shown in Figure 13.

The test bearing used in the experiment is SKF-6202-2Z rolling bearing, and the parameters of the test bearing are given in Table 6. The depth and width of the defect are 0.5 mm and 0.2 mm, respectively. The motor speeds are 1200, 1320, and 1500 rpm, and the motor loads are 0, 2, and 4 N·m. The collected datasets of vibration signals of rolling bearings under various operating conditions are shown in Table 7. In the same way, acquired signals are processed by maximum correlated kurtosis deconvolution to clearly show the presence of faults. Taking rolling bearings under the operating condition of 0 N·m and 1200 rpm as examples, spectrums and envelope spectrums of acquired signals of rolling bearings are respectively shown in Figure 14 and Figure 15. To test the validity of the proposed method in fault identification, the experiments via the EWNAP and BP neural network are conducted here.

Table 6.

Bearing parameters.

Inside diameter, mm	Outside diameter, mm	Ball diameter, mm	Contact angle	Pitch diameter, mm	No. of balls
15	35	6	0°	25	8

Table 7.

Rolling bearing data sets under various operating conditions.

Number of training/testing samples	Type	Load (N·m)	Motor speed (rpm)
20/5	Normal	0	1200
20/5	Normal	2	1320
20/5	Normal	4	1500
20/5	Inner race defect	0	1200
20/5	Inner race defect	2	1320
20/5	Inner race defect	4	1500
20/5	Ball defect	0	1200
20/5	Ball defect	2	1320
20/5	Ball defect	4	1500
20/5	Outer race defect	0	1200
20/5	Outer race defect	2	1320
20/5	Outer race defect	4	1500

The number of points in each sample is 2560. Twenty five samples of each type under one operating condition are obtained in the feature extraction. During EWNAP processing, each type can acquire its respective projection matrix $P_{m}$ , and together they constitute a projection matrix collection $P_{s}$ . These samples must go through the projection with $P_{s}$ before putting into the BP neural network. The sample number of each type is the same: there are 300 sets of normal, inner race defect, ball defect, and outer race defect. The first 60 samples of each type are utilized as training samples and the remaining 15 samples are utilized as testing samples. The structure of BP neural network has been illustrated in Back Propagation Neural Network. The detailed structure and parameters of BP neural network are set as follow: the number of the input-layer neurons is 100, the number of the hidden-layer neurons is 50, the number of the output-layer neurons is 4, the training epoch is 1000, the learning rate is 0.1, and the training accuracy is $4 e^{- 7}$ .²² The labels (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1), denote normal, inner race defect, ball defect, and outer race defect, respectively. The predicted results are presented in Figure 16.

The classification precisions are calculated, which are the percentage of correct classification results. The classification precisions of normal, inner race defect, ball defect, and outer race defect in the testing process of BP neural network are 100%, 100%, 100%, and 100%. To further demonstrate the effectiveness of the proposed method, 15 experiments are conducted to obtain the averages of training accuracy and testing accuracy. The detailed results are displayed in Figure 17.

The average training accuracy and testing accuracy of 15 experiments is 100% and 97.56%, respectively. Thus, the proposed method can be utilized for fault diagnosis under the various operating conditions. To verify the superiority of the proposed method, samples processed by PCA and NAP are used for comparative analysis. The number of samples processed by PCA is 300, where first 80% samples of each type are used for training and the rest of samples are used for testing in BP neural network. The results, as shown in Figure 18, indicate that classification precision for normal, inner race defect, ball defect, and outer race defect is 86.67%, 60%, 60%, and 80%, respectively. The samples are processed by NAP and then put into BP neural network. As shown in Figure 19, the classification precision for normal, inner race defect, ball defect and outer race defect is 93.33%, 100%, 93.33% and 93.33%, respectively.

To further demonstrate the superiority of the proposed fault diagnosis approach, 15 experiments are conducted. Two classical metrics, including recall and precision, are used to evaluate the performance of the above methods. The recall is the percentage of the samples which are recognized successfully and the precision is the accuracy of the results. The average metrics of different methods of the 15 experiments are shown in Figure 20. In terms of recall and precision of different methods, the proposed method achieves the best recall and precision compared with other methods, which demonstrates the superior performance of the proposed fault diagnosis method. The recall and precision of the proposed method are more than 97.5%, which means the proposed approach achieves a high precision while maintaining a high recall. As a result, the samples processed by EWNAP retain more fault information than samples processed by NAP, and fault diagnosis via EWNAP and BP neural network is able to provide more accurate classification results under various operating conditions.

Case 3: Validation of the scheme via EWNAP and BP neural network with insufficient data

To demonstrate the scheme with insufficient data, Case Western Reserve University bearing data and data collected from the bearing fault simulation bench are selected. For Case Western Reserve University bearing data, the sample number of each type under three operating conditions is shown in Table 3, and some samples of each type are selected to train BP neural network. The distribution of training samples under various operating conditions is shown in Table 8. For each type of rolling bearings, 20 training samples are respectively taken under the first two operating conditions. In the third operating condition, 6, 8, 10, 12, and 14 samples are respectively used for training BP neural network. The detailed structure of BP neural network is given as follows, and it has two hidden layers, apart from an input layer and an output layer. Referring to the detailed structure of BP neural network in Li’s paper,⁵⁶ the number of neurons in each layer of BP neural network used in this case is shown in Table 9, and the parameters of BP neural network are set as follow: the training epoch is 1000, the learning rate is 0.1, and the training accuracy is

4 e^{- 7}

. After EWNAP processing, 40 samples of each type under the first two operating conditions are taken to train BP neural network. In the third operating condition, a few samples are regarded as training samples. Comparison experiments are conducted, including using PCA and LPP, and the number of training samples is the same as above. By using 15 testing samples of each type taken from the third operating condition, Figure 21 compares classification accuracies of different methods when the training samples taken from the third operating condition gradually increase.

Table 8.

The number of training samples under various operating conditions.

	Normal	Inner race defect	Ball defect	Outer race defect
Operating condition 1	20	20	20	20
Operating condition 2	20	20	20	20
Operating condition 3	6/8/10/12/14	6/8//1012/14	6/8/10/12/14	6/8/10/12/14

Table 9.

The number of neurons in each layer.

Input layer	Hidden layer (1)	Hidden layer (2)	Output layer
100	50	25	4

From the results shown in Figure 21, it is observed that classification accuracies of samples processed by PCA and LPP are not high when the number of training samples from the third operating condition is few. The training samples are less than 14 and the accuracies are not more than 60%. However, the classification accuracy of samples processed by EWNAP usually remains above 80%, when the number of training samples from the third operating condition is more than 10. Even if the number of training samples from the third operating condition is less than 10, the trained BP neural network can still maintain a certain accuracy. To further demonstrate the proposed scheme, an experiment using the data acquired from bearing fault simulation bench is conducted. As shown in Table 7, the data is also collected under three operating conditions. Thus, the number of training samples is set according to Table 8. Following the same data processing process and the same structure and parameters of BP neural network, the experimental results are shown in Figure 22 by using 11 testing samples of each type taken from the third operating condition. The results are similar to those shown in Figure 21. Compared with the other methods, the classification accuracy of the proposed method keep high when the number of training samples is few. Therefore, the EWNAP combined with BP neural network is better, since only a few training samples from various operating conditions are needed to obtain comparatively more accurate classification results.

Conclusion

In this paper, a new robust fault diagnosis approach via EWNAP and BP neural network is proposed to alleviate the nuisance attributes caused by various operating conditions. The novelties of this paper are multi-fold: first, the fuzzy entropy is employed to quantify the degree of interference in different operating conditions, and then EWNAP is proposed by constructing a new weighted matrix to alleviate the adverse impacts caused by various operating conditions. The processed samples by EWNAP are adopted as the input for BP neural network to achieve fault diagnosis of rolling bearing under various operating conditions. In addition, the proposed fault diagnosis approach shows a better performance with relatively few training samples. The major findings of the proposed approach in this paper can be summarized as follows:

(1) Entropy is employed to quantify the interference degree of various operating conditions, on account of its internal essence of measuring the regularity and complexity of dynamic systems. Fuzzy entropy is used to quantitatively estimate the interference degree of various operating conditions. The novel weighted matrix of NAP is constructed to solve the “bipolar problem” existing in NAP.

(2) The proposed EWNAP scheme by the authors has been validated in alleviating the interference caused by various operating conditions. Such optimization scheme retains primary fault features during the elimination of existing interference and shows superiority compared to original NAP, by proposing a new way to construct the weighted matrix. The case studies validated the superiority of the proposed approach.

(3) The proposed robust approach in this paper shows the effectiveness and superiority in the fault diagnosis of rolling bearings under various operating conditions. Additionally, the effectiveness of the proposed novel scheme with insufficient data has also been validated in this paper. The fault diagnosis of rolling bearings with insufficient data is promising in the industrial practice.

In summary, the proposed robust fault diagnosis approach via EWNAP and BP neural network provides a better solution to fault identification of rotating machinery under various operating conditions. The nuisance attributes are effectively eliminated, which contributes to accurate fault feature extraction without the interference of operating conditions. While EWNAP improves the accuracy of fault identification, it is worth noting that EWNAP is limited because it requires linear space to effectively eliminate interference as same as NAP. In future work, EWNAP will be combined with appropriate feature processing methods to construct a scalable fault diagnosis model, and the effectiveness of EWNAP in multivariate signal processing will be further researched.

Footnotes

Acknowledgements

This research work was supported by National Natural Science Foundation of China under Grant No.51875416, Natural Science Foundation Innovation Group Program of Hubei Province under Grant No.2020CFA033, and China Postdoctoral Science Foundation under Grant No.2020M682492, which are greatly appreciated. The authors appreciate the free download of the original bearing failure data and one photo picture provided by Case Western Reserve University Bearing Data Center Website.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China (51875416), Natural Science Foundation Innovation Group Program of Hubei Province(2020CFA033), and China Postdoctoral Science Foundation(2020M682492).

ORCID iDs

Di Yang

Yong Lv

Rui Yuan

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